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Calculating the area of a polygon can be as simple as finding the area of a regular triangle or as complicated as finding the area of an irregular eleven-sided shape. If you want to know how to find the area of a variety of polygons, just follow these steps.

Write down the formula for finding the area of a regular polygon. To find the area of a regular polygon, all you have to do is follow this simple formula: area = 1/2 x perimeter x apothem.[1]XResearch source Here is what it means:

Perimeter = the sum of the lengths of all the sides

Apothem = a segment that joins the polygon's center to the midpoint of any side that is perpendicular to that side[2]XResearch source

Find the perimeter of the polygon. If the perimeter is provided for you, then you're nearly done, but it's likely that you have a bit more work to do. If the apothem is provided for you and you know that you're working with a regular polygon, then you can use it to find the perimeter. Here's how you do it:

Think of the apothem as being the "x√3" side of a 30-60-90 triangle. You can think of it this way because the hexagon is made up of six equilateral triangles. The apothem cuts one of them in half, creating a triangle with 30-60-90 degree angles.

You know that the side across from the 60 degree angle has length = x√3, the side across from the 30 degree angle has length = x, and the side across from the 90 degree angle has length = 2x. If 10√3 represents "x√3," then you can see that x = 10.

You know that x = half the length of the bottom side of the triangle. Double it to get the full length. The bottom side of the triangle is 20 units long. There are six of these sides to the hexagon, so multiply 20 x 6 to get 120, the perimeter of the hexagon.

Plug the apothem and the perimeter into the formula. If you're using the formula area = 1/2 x perimeter x apothem, then you can plug in 120 for the perimeter and 10√3 for the apothem. Here is what it will look like:

Simplify your answer. You may need to state your answer in decimal instead of square root form. Just use your calculator to find the closest value for √3 and multiply it by 600. √3 x 600 = 1,039.2. This is your final answer.

Find the area of a square. To find the area of a square, just square the length of one side. This is really the same thing as multiplying the base of the square by its height, because the base and height are the same.

Let's say you have a trapezoid with bases that have a length of 6 and 8 and a height of 10. The area is simple [(6 + 8) x 10]/2, which can be simplified to (14 x 10)/2, or 140/2, which makes for an area of 70.

Write down the coordinates of the vertices[3]XResearch source of the irregular polygon. Determining the area for an irregular polygon can be found when you know the coordinates of the vertices.[4]XResearch source

Assuming it's not a square or rectangle, you would have to subdivide the figure into smaller sections consisting of squares, rectangles, triangles, and other shapes whose areas can be easily calculated. Then add those areas together.

Will this formula work for all regular polygons, regardless of the number of sides?

Donagan

Top Answerer

Not sure which formula you mean. Here's a trig formula that will work for any regular polygon if you know the length of a side: A = s²n / [4 tangent(180°/n)], where s is the length of a side, and n is the number of sides. Here's a trig formula that will work if you know the radius of the polygon (distance from the center to any vertex): A = r²n [sine(360°/n)] / 2, where r is the radius, and n is the number of sides. Finally, here's a trig formula that will work if you know the apothem of the polygon (perpendicular distance from the center to any side): A= a²n[tangent(180°/n)], where a is the length of the apothem, and n is the number of sides.

There are two ways: Mentally cut the polygon up into other shapes that are easy to calculate, such as triangles and squares. Then calculate each part and sum up the results. Another approach you might take is to imagine the polygon is a shape that can be easily calculated, such as a square. Then find the area of the square. Next, subtract the area you used to fill the square from the square.

Write down the formula for finding the area of a regular polygon. To find the area of a regular polygon, all you have to do is follow this simple formula: area = 1/2 x perimeter x apothem. Here is what it means: Perimeter = the sum of the lengths of all the sides.

Tips

If you list the points in a clockwise order instead of counterclockwise, you will get the negative of the area. Hence this can be used as a tool to identify the cyclic path or sequence of a given set of points forming a polygon.

This formula computes area with orientation. If you use it on a shape where two of the lines cross like a figure eight, you will get the area surrounded counterclockwise minus the area surrounded clockwise.

References

About This Article

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 51 people, some anonymous, worked to edit and improve it over time. This article has been viewed 1,183,080 times.

To calculate the area of a regular polygon, use the specific formula for that shape. For a triangle, multiply ½ by the length of the base and the height. For a square or rectangle, multiply the length of the base by the height. To find the length of a trapezoid, add the length of the two bases and multiply that number by the height. Then, divide the total by 2 to get the area. Be sure to include units in your answer if they're provided. If you need to calculate the area of an irregularly-shaped polygon, keep reading to learn how!